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Almost sure asymptotic stability of stochastic Volterra integro-differential equations with fading perturbations. (English) Zbl 1121.60070

The authors’ starting point is the deterministic Volterra equation in \(\mathbb{R}^d\) given by \[ \dot{x}(t) = \int_{[0,t]}\nu(ds)x(t-s)\text{ for }t\geq 0\tag{1} \] with initial condition \(x(0)=x_0\) and \(\nu\) a finite measure. It is well known that its zero solution is uniformly asymptotically stable if and only if the resolvent of (1) is in \(L^1(\mathbb{R}_+,\mathbb{R}^{d\times d})\). The resolvent is the unique matrix-valued function solving (1) with the initial condition being the \(d\)-dimensional identity matrix.
The authors first perturb (1) by an additive noise term \(\sigma_1(t) dW(t)\), were \(W\) is a \(d'\)-dimensional Wiener process. They provide conditions on \(\sigma_1\) such that the solution \(X\) of the corresponding stochastic equation satisfies \(\lim_{t\rightarrow \infty} | X(t)| =0\) \(P\)-almost surely, if the resolvent of (1) is in \(L^1(\mathbb{R}_+,\mathbb{R}^{d\times d})\). Then a perturbation of (1) with a multiplicative noise term is considered and the asymptotic stability of the zero solution of the corresponding stochastic equation is investigated.

MSC:

60H20 Stochastic integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34K20 Stability theory of functional-differential equations
45D05 Volterra integral equations
93D20 Asymptotic stability in control theory
93E15 Stochastic stability in control theory
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References:

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